Slow Draining Bathroom Sink

I currently have one bathroom sink that is draining slowly. I took apart the s-trap and there is no clog there or around the stopper. I believe it maybe a venting problem because the sink backs up almost immediatly after the water is turned on. If I losen the pvc connetion after the s-trap that connects it to the in floor drain pipe the water in the sink drains faster, however the connection starts to leak.

No other fixture in the house has this drainage problem. This particular sink drains into a main drain pipe that other fixture connect to and the vent pipe goes through the roof, so I am not sure what the problem is.

Could the run from this fixture to the main drain be partially blocked causing a venting problem? as I said previously there is no drainage problem after the trap, but could a partially blocked run to the main drain cause some kind of pressure issue in the trap?

Is this sink a vessel sink or one without an overflow? Does it have a grid drain , or is it a pop-up stopper? It's unlikely that it is a venting issue. It could be a partial clog. If the sink does not have an overflow, those can be notorious for not draining well depending on the type of drain cover it has.

It is a pop-up stopper. And it does have an overflow. The old sink that was there previously also had a draining problem with no clog (at least not from the sink to the vertical drain line. The reason I think it is a venting problem is because the drain backs up no more then 3 seconds after the water is turned on and there is a good 4 feet of vertical pipe after the s-trap. I even tried taking the s-trap off and pouring water down the vertical drain pipe and there was no issue.

For some reason water is not flowing freely past the s-trap even though there is no blockage.

Ok, here is what was found. After the trap connects to the vertical drain pipe this leads to our laundry room which connects to a 90 degree elbow to a parallel to the ceiling pipe to the main sewage line. This parallel run is the "old plumbing" of the house. In this parallel run the pipe has caked on "soil" almost like the line was backed up at one time with soot before we moved in. It looks like a clogged artery leaving the hole in this parallel run about an inch wide. I guess this is the reason why this particular fixture is having a venting problem.